In the 3-4-13 germanides family, Lu\(_{3}\) Os\(_{4}\) Ge\(_{13}\) , a semimetallic compound, has shown interesting properties in the superconducting state and attracted attention recently. Two initial studies of Lu\(_{3}\) Os\(_{4}\) Ge\(_{13}\)  by Prakash et al. revealed two-gap bulk superconductivity from low-temperature electronic specific heat measurements and a non-linear dependence of the Sommerfeld coefficient, \(\gamma _n\) , under magnetic field [1]}, [2]}. The superfluid density measurements via tunnel diode oscillator (TDO) also confirmed the two-gap nature of the superconducting state [3]}. Another noteworthy property of Lu\(_{3}\) Os\(_{4}\) Ge\(_{13}\)  is the high value of the upper critical field, \(H_{c2}\) , which is close to its Pauli limiting value [4]}, and the high transition temperature \(T_c\)  for a relatively low carrier density system. Other low carrier density systems with high \(T_c\)  include cuprates, fullerenes and MgB\(_2\) , which are known for their fascinating properties [5]}, [6]}, [7]}.
Exploring the phenomenology of low-carrier-density superconductors is also crucial for our theoretical understanding as the conventional BCS theory is not applicable to the low-density limit.
This motivates a detailed microscopic investigation of the germanide superconductor Lu\(_{3}\) Os\(_{4}\) Ge\(_{13}\)  in order to understand its superconducting order parameter, the pairing mechanism and their relation with the structure in the superconducting phase.
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